Frequentist inference

Frequentist inference is a type of statistical inference based in frequentist probability, which treats “probability” as equivalent with “frequency” and draws conclusions from sample data by emphasising the frequency or proportion of findings in the data. The frequentist inference underlies frequentist statistics, in which the well-established methodologies of statistical hypothesis testing and confidence intervals are based.

Experimental design and methodology

Frequentist inferences are associated with the application frequentist probability to experimental design and interpretation, and specifically with the view that any given experiment can be considered one of an infinite sequence of possible repetitions of the same experiment, each capable of producing statistically independent results. In this view, the frequentist inference approach to drawing conclusions from data is effectively to require that the correct conclusion should be drawn with a given (high) probability, among this notional set of repetitions.

However, exactly the same procedures can be developed under a subtly different formulation. This is one where a pre-experiment point of view is taken. It can be argued that the design of an experiment should include, before undertaking the experiment, decisions about exactly what steps will be taken to reach a conclusion from the data yet to be obtained. These steps can be specified by the scientist so that there is a high probability of reaching a correct decision where, in this case, the probability relates to a yet to occur set of random events and hence does not rely on the frequency interpretation of probability. This formulation has been discussed by Neyman, among others.

Relationship with other approaches

Main article: Statistical inference § Paradigms for inference

Further information: Probability interpretations

Frequentist inferences stand in contrast to other types of statistical inferences, such as Bayesian inferences and fiducial inferences. While the "Bayesian inference" is sometimes held to include the approach to inferences leading to optimal decisions, a more restricted view is taken here for simplicity.

Bayesian inference

Main article: Bayesian inference § In frequentist statistics and decision theory

Bayesian inference is based in Bayesian probability, which treats “probability” as equivalent with “certainty”, and thus that the essential difference between the frequentist inference and the Bayesian inference is the same as the difference between the two interpretations of what a “probability” means. However, where appropriate, Bayesian inferences (meaning in this case an application of Bayes' theorem) are used by those employing frequency probability.

There are two major differences in the frequentist and Bayesian approaches to inference that are not included in the above consideration of the interpretation of probability:

1. In a frequentist approach to inference, unknown parameters are often, but not always, treated as having fixed but unknown values that are not capable of being treated as random variates in any sense, and hence there is no way that probabilities can be associated with them. In contrast, a Bayesian approach to inference does allow probabilities to be associated with unknown parameters, where these probabilities can sometimes have a frequency probability interpretation as well as a Bayesian one. The Bayesian approach allows these probabilities to have an interpretation as representing the scientist's belief that given values of the parameter are true (see Bayesian probability - Personal probabilities and objective methods for constructing priors).
2. While "probabilities" are involved in both approaches to inference, the probabilities are associated with different types of things. The result of a Bayesian approach can be a probability distribution for what is known about the parameters given the results of the experiment or study. The result of a frequentist approach is either a "true or false" conclusion from a significance test or a conclusion in the form that a given sample-derived confidence interval covers the true value: either of these conclusions has a given probability of being correct, where this probability has either a frequency probability interpretation or a pre-experiment interpretation.

See also

• Intuitive statistics
• German tank problem

Bibliography

• Everitt, B.S. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press. ISBN 0-521-81099-X.
• Jerzy, Neyman (1937). "Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability". Philosophical Transactions of the Royal Society of London A: 236, 333–380.